Learn Hardcore Turkish: Ben kütüphaneye doğru yürüyorum.

Because you need the dative case to show motion toward something. The dative suffix in Turkish is -e (or -a), but when the noun ends in a vowel you insert a buffer y. Since kütüphane ends in -e, you add -ye, giving kütüphaneye (‘to the library’ or ‘towards the library’).

What role does doğru play? I thought doğru meant ‘correct’.

In isolation doğru can indeed mean ‘correct’ or ‘straight’. But when you put it after a noun in the dative case (noun + doğru), it means ‘towards’ or ‘in the direction of’. So kütüphaneye doğru literally means ‘towards the library’.

Can I drop doğru and just say kütüphaneye yürüyorum?

Yes, kütüphaneye yürüyorum (‘I am walking to the library’) is perfectly fine. The version with doğru emphasizes the direction itself rather than stating the library is your ultimate destination.

What are the parts of yürüyorum?

yürüyorum breaks down into three pieces:

yürü- (verb stem, ‘walk’)
-yor- (present continuous marker)
-um (first-person singular ending)
Put together, they mean ‘I am walking.’

Why is Ben included at the start? Is it necessary?

Turkish is a pro-drop (null-subject) language: the verb ending -um already tells you the subject is ‘I’. Ben (‘I’) is optional and is used for emphasis or clarity. If you prefer brevity, you can simply say Kütüphaneye doğru yürüyorum.

Why is the order Ben kütüphaneye doğru yürüyorum (S + direction + V) instead of the English S-V-O order?

Turkish normally follows Subject–Object–Verb or Subject–[Adjunct/Direction]–Verb. Directional phrases like kütüphaneye doğru typically come before the verb, and the verb comes last.

What’s the difference between yürüyorum and gidiyorum? Both mean ‘I’m going’, right?

gidiyorum means ‘I am going’ in general—it doesn’t specify how you go. yürüyorum specifies that you are walking. Use yürüyorum when you want to say ‘I’m walking’, and gidiyorum when mode of travel isn’t important or is implied otherwise.